Impact of Macroeconomic Factors on Amazon Stock Price

The Consumer Discretionary sector comprises companies that produce non-essential goods and services, such as apparel, entertainment, and luxury items. One of the top companies in this sector is Amazon.com, Inc. (AMZN), which has revolutionized the retail industry by providing customers with an online platform to purchase a wide range of products.

By using an ARIMAX+ARCH/GARCH model to analyze the stock price behavior of Amazon, we can gain insights into how macroeconomic factors impact the company’s performance. For example, an increase in GDP growth rate or a decrease in unemployment rate may lead to increased consumer spending and a rise in Amazon’s stock price. Conversely, an increase in inflation or interest rates may lead to a decrease in consumer spending and a decline in Amazon’s stock price.

Time series Plot

Amazon

Code

# get data
options("getSymbols.warning4.0"=FALSE)
options("getSymbols.yahoo.warning"=FALSE)


data.info = getSymbols("AMZN",src='yahoo', from = '2010-01-01',to = "2023-03-01",auto.assign = FALSE)
data = getSymbols("AMZN",src='yahoo', from = '2010-01-01',to = "2023-03-01")
df <- data.frame(Date=index(AMZN),coredata(AMZN))

# create Bollinger Bands
bbands <- BBands(AMZN[,c("AMZN.High","AMZN.Low","AMZN.Close")])

# join and subset data
df <- subset(cbind(df, data.frame(bbands[,1:3])), Date >= "2010-01-01")


# colors column for increasing and decreasing
for (i in 1:length(df[,1])) {
  if (df$AMZN.Close[i] >= df$AMZN.Open[i]) {
      df$direction[i] = 'Increasing'
  } else {
      df$direction[i] = 'Decreasing'
  }
}

i <- list(line = list(color = '#7FB3D5'))
d <- list(line = list(color = '#7F7F7F'))

# plot candlestick chart

fig <- df %>% plot_ly(x = ~Date, type="candlestick",
          open = ~AMZN.Open, close = ~AMZN.Close,
          high = ~AMZN.High, low = ~AMZN.Low, name = "AMZN",
          increasing = i, decreasing = d) 
fig <- fig %>% add_lines(x = ~Date, y = ~up , name = "B Bands",
            line = list(color = '#ccc', width = 0.5),
            legendgroup = "Bollinger Bands",
            hoverinfo = "none", inherit = F) 
fig <- fig %>% add_lines(x = ~Date, y = ~dn, name = "B Bands",
            line = list(color = '#ccc', width = 0.5),
            legendgroup = "Bollinger Bands", inherit = F,
            showlegend = FALSE, hoverinfo = "none") 
fig <- fig %>% add_lines(x = ~Date, y = ~mavg, name = "Mv Avg",
            line = list(color = '#C052B3', width = 0.5),
            hoverinfo = "none", inherit = F) 
fig <- fig %>% layout(yaxis = list(title = "Price"))

# plot volume bar chart
fig2 <- df 
fig2 <- fig2 %>% plot_ly(x=~Date, y=~AMZN.Volume, type='bar', name = "AMZN Volume",
          color = ~direction, colors = c('#7FB3D5','#7F7F7F')) 
fig2 <- fig2 %>% layout(yaxis = list(title = "Volume"))

# create rangeselector buttons
rs <- list(visible = TRUE, x = 0.5, y = -0.055,
           xanchor = 'center', yref = 'paper',
           font = list(size = 9),
           buttons = list(
             list(count=1,
                  label='RESET',
                  step='all'),
             list(count=3,
                  label='3 YR',
                  step='year',
                  stepmode='backward'),
             list(count=1,
                  label='1 YR',
                  step='year',
                  stepmode='backward'),
             list(count=1,
                  label='1 MO',
                  step='month',
                  stepmode='backward')
           ))

# subplot with shared x axis
fig <- subplot(fig, fig2, heights = c(0.7,0.2), nrows=2,
             shareX = TRUE, titleY = TRUE)
fig <- fig %>% layout(title = paste("Amazon Stock Price: January 2010 - March 2023"),
         xaxis = list(rangeselector = rs),
         legend = list(orientation = 'h', x = 0.5, y = 1,
                       xanchor = 'center', yref = 'paper',
                       font = list(size = 10),
                       bgcolor = 'transparent'))

fig

Differentitaed Chart Series

Code

log(data.info$`AMZN.Adjusted`) %>% diff() %>% chartSeries(theme=chartTheme('white'),up.col='#7FB3D5')

GDP Growth

Code

#import the data
gdp <- read.csv("DATA/RAW DATA/gdp-growth.csv")

#change date format
gdp$Date <- as.Date(gdp$DATE , "%m/%d/%Y")

#drop DATE column
gdp <- subset(gdp, select = -c(1))

#export the cleaned data
gdp_clean <- gdp
write.csv(gdp_clean, "DATA/CLEANED DATA/gdp_clean_data.csv", row.names=FALSE)

#plot gdp growth rate 
fig <- plot_ly(gdp, x = ~Date, y = ~value, type = 'scatter', mode = 'lines',line = list(color = 'rgb(240, 128, 128)'))
fig <- fig %>% layout(title = "U.S GPD Growth Rate: 2010 - 2022",xaxis = list(title = "Time"),yaxis = list(title ="GDP Growth Rate"))
fig

Inflation

Code

#import the data
inflation_rate <- read.csv("DATA/RAW DATA/inflation-rate.csv")

#cleaning the data
#remove unwanted columns
inflation_rate_clean <- subset(inflation_rate, select = -c(1,HALF1,HALF2))

#convert the data to time series data
inflation_data_ts <- ts(as.vector(t(as.matrix(inflation_rate_clean))), start=c(2010,1), end=c(2023,2), frequency=12)

#export the data
write.csv(inflation_rate_clean, "DATA/CLEANED DATA/inflation_rate_clean_data.csv", row.names=FALSE)


#plot inflation rate 
fig <- autoplot(inflation_data_ts, ylab = "Inflation Rate", color="#FFA07A")+ggtitle("U.S Inflation Rate: January 2010 - February 2023")+theme_bw()
ggplotly(fig)

Interest

Code

#import the data
interest_data <- read.csv("DATA/RAW DATA/interest-rate.csv")

#change date format
interest_data$Date <- as.Date(interest_data$Date , "%m/%d/%Y")

#export the cleaned data
interest_clean_data <- interest_data
write.csv(interest_clean_data, "DATA/CLEANED DATA/interest_rate_clean_data.csv", row.names=FALSE)

#plot interest rate 
fig <- plot_ly(interest_data, x = ~Date, y = ~value, type = 'scatter', mode = 'lines',line = list(color='rgb(219, 112, 147)'))
fig <- fig %>% layout(title = "U.S Interest Rate: January 2010 - March 2023",xaxis = list(title = "Time"),yaxis = list(title ="Interest Rate"))
fig

Unemployment

Code

#import the data
unemployment_rate <- read.csv("DATA/RAW DATA/unemployment-rate.csv")

#change date format
unemployment_rate$Date <- as.Date(unemployment_rate$Date , "%m/%d/%Y")

# export the data
write.csv(unemployment_rate, "DATA/CLEANED DATA/unemployment_rate_clean_data.csv", row.names=FALSE)

#plot unemployment rate 
fig <- plot_ly(unemployment_rate, x = ~Date, y = ~Value, type = 'scatter', mode = 'lines',line = list(color = 'rgb(189, 183, 107)'))
fig <- fig %>% layout(title = "U.S Unemployment Rate: January 2010 - March 2023",xaxis = list(title = "Time"),yaxis = list(title ="Unemployment Rate"))
fig

From 2010 to 2023, Amazon’s stock experienced significant growth as the company expanded its operations and diversified its offerings. In 2010, Amazon was primarily known as an online retailer of books, electronics, and other consumer goods. However, over the next several years, the company expanded into new markets, including streaming video and music, cloud computing services, and even physical retail stores.

Throughout this period, Amazon’s stock price saw consistent growth, although there were occasional dips in response to macroeconomic events or company-specific news. For example, the company’s stock price declined in 2014 amid concerns about its profitability and increased competition in the retail industry. However, Amazon’s stock quickly rebounded as the company continued to innovate and expand into new markets.

In recent years, Amazon’s stock price has been impacted by a variety of factors, including the COVID-19 pandemic and increased scrutiny from regulators. Nevertheless, the company’s strong position in the e-commerce and cloud computing markets, as well as its continued investment in new technologies and initiatives, have helped to maintain investor confidence in Amazon’s long-term growth potential.

Since early 2021, Amazon’s stock price has experienced some volatility, likely due to a combination of factors such as global economic uncertainty and fluctuations in consumer demand for the company’s products.

As discussed before, the macroeconomic factors of GDP growth, inflation, interest rates, and unemployment rate are closely interrelated and play a crucial role in the overall health and stability of an economy. From 2010 to 2023, the global economy experienced a mix of ups and downs, with periods of strong GDP growth followed by slowdowns and recessions.

The second plot shows the first difference of the logarithm of the adjusted Amazon stock price. Taking the first difference removes any long-term trends and transforms the time series into a stationary process. From the plot, we can observe that the first difference of the logarithm of the Amazon stock price appears to be stationary, as the mean and variance are roughly constant over time.

Enodogenous and Exogenous Variables

Plot

Code

numeric_data <- c("AMZN.Adjusted","gdp", "interest", "inflation", "unemployment")
numeric_data <- final[, numeric_data]
normalized_data_numeric <- scale(numeric_data)
normalized_data <- ts(normalized_data_numeric, start = c(2010, 1), end = c(2021,10),frequency = 4)
ts_plot(normalized_data,
        title = "Normalized Time Series Data for AMZN Stock and Macroeconomic Variables",
        Ytitle = "Normalized Values",
        Xtitle = "Year")

Correlation Heatmap

Code

# Get upper triangle of the correlation matrix
get_upper_tri <- function(cormat){
    cormat[lower.tri(cormat)]<- NA
    return(cormat)
}
cormat <- round(cor(normalized_data_numeric),2)

upper_tri <- get_upper_tri(cormat)

melted_cormat <- melt(upper_tri, na.rm = TRUE)
# Create a ggheatmap
ggheatmap <- ggplot(melted_cormat, aes(Var2, Var1, fill = value))+
 geom_tile(color = "white")+
 scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
   midpoint = 0, limit = c(-1,1), space = "Lab", 
    name="Pearson\nCorrelation") +
  theme_minimal()+ # minimal theme
 theme(axis.text.x = element_text(angle = 45, vjust = 1, 
    size = 12, hjust = 1))+
 coord_fixed()

ggheatmap + 
geom_text(aes(Var2, Var1, label = value), color = "black", size = 4) +
theme(
  axis.title.x = element_blank(),
  axis.title.y = element_blank(),
  panel.grid.major = element_blank(),
  panel.border = element_blank(),
  panel.background = element_blank(),
  axis.ticks = element_blank(),
  legend.justification = c(1, 0),
  legend.position = c(0.6, 0.7),
  legend.direction = "horizontal")+
  guides(fill = guide_colorbar(barwidth = 7, barheight = 1,
                title.position = "top", title.hjust = 0.5))

CCF GDP

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AMZN.Adjusted")], normalized_data[, c("gdp")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AMZN Stock Price and GDP Growth Rate ",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 4.585922

CCF Interest

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AMZN.Adjusted")], normalized_data[, c("interest")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AMZN Stock Price and Interest Rate",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 11.29588

CCF Inflation

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AMZN.Adjusted")], normalized_data[, c("inflation")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AMZN Stock Price and Inflation Rate",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 16.68466

CCF Unemployment

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AMZN.Adjusted")], normalized_data[, c("unemployment")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AMZN Stock Priceand Unemployment Rate",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 20.53876

The Normalized Time Series Data for Stock Price and Macroeconomic Variables plot shows the same variables as the first plot but has been normalized to a common range of 0 to 1 using the scale() function in R, which standardizes the variables to have a mean of 0 and a standard deviation of 1. The heatmap analysis of the normalized data reveals that inflation and unemployment rate exhibit strong positive correlations with the stock price indices, indicating that these variables may significantly influence stock price movements. On the other hand, weaker correlations were observed between the stock price indices and GDP and interest rates, suggesting that these variables may have less impact on stock price fluctuations. The cross-correlation feature plots confirm these findings, indicating that inflation and unemployment rate are more suitable feature variables for the ARIMAX model when predicting Amazon movements.

Final Exogenous variables: Macroeconomic indicators: Inflation rate and unemployment rate.

Enodogenous and Exogenous Variables Plot

Plot

Code

final_data <- final %>%dplyr::select( Date,AMZN.Adjusted, inflation,unemployment)
numeric_data <- c("AMZN.Adjusted", "inflation","unemployment")
numeric_data <- final_data[, numeric_data]
normalized_data_numeric <- scale(numeric_data)
normalized_numeric_df <- data.frame(normalized_data_numeric)
normalized_data_ts <- ts(normalized_data_numeric, start = c(2010, 1), frequency = 4)

autoplot(normalized_data_ts, facets=TRUE) +
  xlab("Year") + ylab("") +
  ggtitle("Amazon Stock Price, Inflation Rate and Unemployment Rate in USA 2010-2023")

Check the stationarity

Code

# Convert your multivariate time series data to a matrix
final_data_ts_multivariate <- as.matrix(normalized_data_ts)

# Check for stationarity using Phillips-Perron test
phillips_perron_test <- ur.pp(final_data_ts_multivariate)  
summary(phillips_perron_test)

################################## 
# Phillips-Perron Unit Root Test # 
################################## 

Test regression with intercept 


Call:
lm(formula = y ~ y.l1)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.4360 -0.1440 -0.0602  0.0993  4.5029 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.0001847  0.0403454   0.005    0.996    
y.l1        0.8645022  0.0407542  21.213   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5023 on 153 degrees of freedom
Multiple R-squared:  0.7463,    Adjusted R-squared:  0.7446 
F-statistic:   450 on 1 and 153 DF,  p-value: < 2.2e-16


Value of test-statistic, type: Z-alpha  is: -22.0542 

         aux. Z statistics
Z-tau-mu            0.0055

The results of the Phillips-Perron unit root test indicate strong evidence against the null hypothesis of a unit root, as the p-value for the coefficient of the lagged variable is less than the significance level of 0.05. This suggests that the variable y, which is being tested for stationarity, is likely stationary. Furthermore, the test statistic Z-tau-mu is 0.0055, which is smaller than the critical value of Z-alpha (-22.0542), providing further evidence of stationarity.

To determine whether the linear model requires an ARCH model, an ARCH test is conducted. The ACF and PACF plots are also used to identify suitable model values.

Model Fitting

Plot

Code

normalized_numeric_df$AMZN.Adjusted<-ts(normalized_numeric_df$AMZN.Adjusted,star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
normalized_numeric_df$inflation<-ts(normalized_numeric_df$inflation,star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
normalized_numeric_df$unemployment<-ts(normalized_numeric_df$unemployment,star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)

fit <- lm(AMZN.Adjusted ~ inflation+unemployment, data=normalized_numeric_df)
fit.res<-ts(residuals(fit),star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
############## Then look at the residuals ############
returns <- fit.res  %>% diff()
autoplot(returns)+ggtitle("Linear Model Returns")

ARCH Test

Code

byd.archTest <- ArchTest(fit.res, lags = 1, demean = TRUE)
byd.archTest

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  fit.res
Chi-squared = 36.896, df = 1, p-value = 1.246e-09

ACF Plot

Code

ggAcf(returns) +ggtitle("ACF for returns")

PACF Plot

Code

ggPacf(returns) +ggtitle("PACF for returns")

The ARCH LM-test was conducted with the null hypothesis of no ARCH effects. The test resulted in a chi-squared value of 36.896 with one degree of freedom, and a very low p-value of 1.246e-09. This suggests strong evidence against the null hypothesis, indicating the presence of ARCH effects in the data.

Based on the ACF and PACF plots, it appears that there is some significant autocorrelation and partial autocorrelation at multiple lags, which suggests that an ARIMA model may not be sufficient to capture the time series behavior. Additionally, the values for p and q appear to be relatively high, with p = 1 and q = 1 being suggested by the plots.

ARIMAX Model

Auto Arima Model

Code

xreg <- cbind(Inflation = normalized_data_ts[, "inflation"],
              Unemployment = normalized_data_ts[, "unemployment"])
fit.auto <- auto.arima(normalized_data_ts[, "AMZN.Adjusted"], xreg = xreg)
summary(fit.auto)

Series: normalized_data_ts[, "AMZN.Adjusted"] 
Regression with ARIMA(0,1,0) errors 

Coefficients:
      Inflation  Unemployment
         0.0625       -0.0687
s.e.     0.0966        0.0438

sigma^2 = 0.04933:  log likelihood = 5.39
AIC=-4.78   AICc=-4.27   BIC=1.01

Training set error measures:
                     ME     RMSE       MAE      MPE     MAPE     MASE      ACF1
Training set 0.03057732 0.215598 0.1225672 11.69566 31.46174 0.408043 0.1470963

Auto Arima Residuals

Code

checkresiduals(fit.auto)

    Ljung-Box test

data:  Residuals from Regression with ARIMA(0,1,0) errors
Q* = 4.4858, df = 8, p-value = 0.8109

Model df: 0.   Total lags used: 8

ARIMA (1,1,1) Plot

Code

set.seed(1234)

model_output <- capture.output(sarima(fit.res, 1,1,1)) 

ARIMA (1,1,1) Model

Code

cat(model_output[30:61], model_output[length(model_output)], sep = "\n")

Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), 
    xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, 
        REPORT = 1, reltol = tol))

Coefficients:
         ar1      ma1  constant
      0.6076  -0.2899   -0.0064
s.e.  0.2341   0.2628    0.0647

sigma^2 estimated as 0.06782:  log likelihood = -3.83,  aic = 15.66

$degrees_of_freedom
[1] 48

$ttable
         Estimate     SE t.value p.value
ar1        0.6076 0.2341  2.5960  0.0125
ma1       -0.2899 0.2628 -1.1031  0.2755
constant  -0.0064 0.0647 -0.0984  0.9220

$AIC
[1] 0.3070239

$AICc
[1] 0.3170364

$BIC
[1] 0.4585396

NA
NA

Cross Validation

Code

n=length(fit.res)
k= 51
 
 
rmse1 <- matrix(NA, (n-k),4)
rmse2 <- matrix(NA, (n-k),4)
rmse3 <- matrix(NA, (n-k),4)


st <- tsp(fit.res)[1]+(k-5)/4 

for(i in 1:(n-k))
{
  xtrain <- window(fit.res, end=st + i/4)
  xtest <- window(fit.res, start=st + (i+1)/4, end=st + (i+4)/4)
  
  #ARIMA(0,1,0) ARIMA(1,1,1)
  
  fit <- Arima(xtrain, order=c(0,1,0),
                include.drift=TRUE, method="ML")
  fcast <- forecast(fit, h=4)
  
  fit2 <- Arima(xtrain, order=c(1,1,1),
                include.drift=TRUE, method="ML")
  fcast2 <- forecast(fit2, h=4)


  rmse1[i,1:length(xtest)]   <- sqrt((fcast$mean-xtest)^2)
  rmse2[i,1:length(xtest)] <- sqrt((fcast2$mean-xtest)^2)
}

plot(1:4,colMeans(rmse1,na.rm=TRUE), type="l",col=2, xlab="horizon", ylab="RMSE")
lines(1:4, colMeans(rmse2,na.rm=TRUE), type="l",col=3)
legend("topleft",legend=c("fit1","fit2"),col=2:4,lty=1)

Based on the results of the auto.arima function, the suggested best model is ARIMA(0,1,0), but the acf and pacf plots suggest a simpler ARIMA(1,1,1) model. To determine the best model, we conduct cross-validation and compare the RMSE values of both models. The results show that ARIMA(1,1,1) has lower RMSE values than ARIMA(0,1,0), indicating that it is the better model.

We can then proceed to choose the best GARCH model using ARIMA(1,1,1) as the base model.

Squared Residuals

Plot

Code

fit <- lm(AMZN.Adjusted ~ inflation+unemployment, data=normalized_numeric_df)
fit.res<-ts(residuals(fit),star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
fit <- Arima(fit.res,order=c(1,1,1))
res=fit$res
plot(res^2,main='Squared Residuals')

ACF Plot

Code

acf(res^2,24, main = "ACF Residuals Square")

PACF Plot

Code

pacf(res^2,24, main = "PACF Residuals Square")

GRACH(1,1)

Code

summary(garchFit(~garch(1,1),res, trace=F))

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~garch(1, 1), data = res, trace = F) 

Mean and Variance Equation:
 data ~ garch(1, 1)
<environment: 0x1557b63a0>
 [data = res]

Conditional Distribution:
 norm 

Coefficient(s):
         mu        omega       alpha1        beta1  
-0.00530411   0.00068953   0.48627960   0.65780719  

Std. Errors:
 based on Hessian 

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)    
mu     -0.0053041   0.0155220   -0.342   0.7326    
omega   0.0006895   0.0011861    0.581   0.5610    
alpha1  0.4862796   0.2392140    2.033   0.0421 *  
beta1   0.6578072   0.1208067    5.445 5.18e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 9.169268    normalized:  0.1763321 

Description:
 Tue Jan  9 20:58:23 2024 by user:  


Standardised Residuals Tests:
                                Statistic p-Value   
 Jarque-Bera Test   R    Chi^2  1.08557   0.5811275 
 Shapiro-Wilk Test  R    W      0.9616717 0.09248575
 Ljung-Box Test     R    Q(10)  5.843556  0.8282312 
 Ljung-Box Test     R    Q(15)  8.260042  0.9129345 
 Ljung-Box Test     R    Q(20)  15.71232  0.7343031 
 Ljung-Box Test     R^2  Q(10)  9.933015  0.4463896 
 Ljung-Box Test     R^2  Q(15)  11.22686  0.7363534 
 Ljung-Box Test     R^2  Q(20)  13.8952   0.8357696 
 LM Arch Test       R    TR^2   13.33945  0.3448583 

Information Criterion Statistics:
        AIC         BIC         SIC        HQIC 
-0.19881801 -0.04872234 -0.20956333 -0.14127488 

From the squared residuals of the best ARIMA model, it can be observed that the ACF plot and PACF plot indicate that the residuals are not autocorrelated and are white noise, indicating a good fit of the model. Based on the squared residuals of the best ARIMA model, we can see that the ACF and PACF plots indicate that most of the values lie between the blue lines. Additionally, the p-value is 1 and q-value is 1. This suggests that the model has a good fit and that there is no significant autocorrelation or partial autocorrelation in the residuals.

The bst model is ARIMA(1,1,1) and GARCH(1,1). #### Best Model

ARIMA Model

Code

#fiting an ARIMA model to the Inflation variable
inflation_fit<-auto.arima(normalized_numeric_df$inflation) 
finflation<-forecast(inflation_fit)

#fitting an ARIMA model to the Unemployment variable
unemployment_fit<-auto.arima(normalized_numeric_df$unemployment) 
funemployment<-forecast(unemployment_fit)

# best model fit for forcasting
xreg <- cbind(Inflation = normalized_data_ts[, "inflation"],
              Unemployment = normalized_data_ts[, "unemployment"])



summary(arima.fit<-Arima(normalized_data_ts[, "AMZN.Adjusted"],order=c(1,1,1),xreg=xreg),include.drift = TRUE)

Series: normalized_data_ts[, "AMZN.Adjusted"] 
Regression with ARIMA(1,1,1) errors 

Coefficients:
          ar1     ma1  Inflation  Unemployment
      -0.5637  1.0000    -0.0025       -0.0790
s.e.   0.1390  0.0573     0.0949        0.0336

sigma^2 = 0.0412:  log likelihood = 9.68
AIC=-9.36   AICc=-8.03   BIC=0.3

Training set error measures:
                     ME      RMSE      MAE      MPE     MAPE      MASE
Training set 0.02911543 0.1929808 0.114944 11.94347 27.98997 0.3826642
                    ACF1
Training set -0.06750725

GARCH Model

Code

summary(final.fit <- garchFit(~garch(1,1), res,trace = F))

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~garch(1, 1), data = res, trace = F) 

Mean and Variance Equation:
 data ~ garch(1, 1)
<environment: 0x15548d140>
 [data = res]

Conditional Distribution:
 norm 

Coefficient(s):
         mu        omega       alpha1        beta1  
-0.00530411   0.00068953   0.48627960   0.65780719  

Std. Errors:
 based on Hessian 

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)    
mu     -0.0053041   0.0155220   -0.342   0.7326    
omega   0.0006895   0.0011861    0.581   0.5610    
alpha1  0.4862796   0.2392140    2.033   0.0421 *  
beta1   0.6578072   0.1208067    5.445 5.18e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 9.169268    normalized:  0.1763321 

Description:
 Tue Jan  9 20:58:24 2024 by user:  


Standardised Residuals Tests:
                                Statistic p-Value   
 Jarque-Bera Test   R    Chi^2  1.08557   0.5811275 
 Shapiro-Wilk Test  R    W      0.9616717 0.09248575
 Ljung-Box Test     R    Q(10)  5.843556  0.8282312 
 Ljung-Box Test     R    Q(15)  8.260042  0.9129345 
 Ljung-Box Test     R    Q(20)  15.71232  0.7343031 
 Ljung-Box Test     R^2  Q(10)  9.933015  0.4463896 
 Ljung-Box Test     R^2  Q(15)  11.22686  0.7363534 
 Ljung-Box Test     R^2  Q(20)  13.8952   0.8357696 
 LM Arch Test       R    TR^2   13.33945  0.3448583 

Information Criterion Statistics:
        AIC         BIC         SIC        HQIC 
-0.19881801 -0.04872234 -0.20956333 -0.14127488 

Volatility

Code

ht <- final.fit@h.t #a numeric vector with the conditional variances (h.t = sigma.t^delta)

#############################
data=data.frame(final)
data$Date<-as.Date(data$Date,"%Y-%m-%d")


data2= data.frame(ht,data$Date)
ggplot(data2, aes(y = ht, x = data.Date)) + geom_line(col = '#7FB3D5') + ylab('Conditional Variance') + xlab('Date')

From the ARIMA(1,1,1), we see that the training set error measures also suggest a good fit, with low mean absolute error, root mean squared error, and autocorrelation of the residuals. GATCH(1,1) model model is used to estimate the volatility of the standardized residuals of the previous regression model. The model includes a mean equation that estimates the mean of the residuals and a variance equation that models the conditional variance of the residuals. The coefficients of the mean equation suggest that the mean of the residuals is close to zero. The variance equation coefficients suggest that the conditional variance of the residuals is dependent on the past conditional variances and the past squared standardized residuals. The model’s the AIC, BIC, SIC, and HQIC values are all relatively low, indicating a good fit of the model. The standardized residuals tests indicate that the residuals are approximately normally distributed and that there is no significant autocorrelation in the residuals.

The volatility of the model seems high in 2020 but has decreased gradually in the past few months. This could indicate that the asset’s price was experiencing a lot of fluctuations in 2020, but the market has stabilized recently.

Model Diagnostics

Residuals

Code

fit2<-garch(res,order=c(1,1),trace=F)
checkresiduals(fit2) 

QQ Plot

Code

qqnorm(fit2$residuals, pch = 1)
qqline(fit2$residuals, col = "blue", lwd = 2)

Box Test

Code

Box.test (fit2$residuals, type = "Ljung")

    Box-Ljung test

data:  fit2$residuals
X-squared = 2.0187, df = 1, p-value = 0.1554

The ACF plot of the residuals shows all the values between the blue lines, which indicates that the residuals are not significantly autocorrelated. The range of values for the residual plot between -2 and 2 is considered acceptable. Additionally, the QQ plot of the residuals shows a linear plot on the line, which is another good indication that the residuals are normally distributed. The QQ plot is a valuable tool to assess if the residuals follow a normal distribution, and in this case, the plot suggests that the residuals do indeed follow a normal distribution.

The Box-Ljung test, a p-value of 0.1554 indicates that the model’s residuals are not significantly autocorrelated, meaning that the model has captured most of the information in the data. This result is good because it suggests that the model is a good fit for the data and has accounted for most of the underlying patterns in the data. Therefore, we can rely on the model’s predictions and use them to make informed decisions.

Forecast

Code

predict(final.fit, n.ahead = 5, plot=TRUE)

  meanForecast meanError standardDeviation lowerInterval upperInterval
1 -0.005304111 0.5028875         0.5028875    -0.9909454     0.9803372
2 -0.005304111 0.5385390         0.5385390    -1.0608212     1.0502130
3 -0.005304111 0.5766303         0.5766303    -1.1354787     1.1248705
4 -0.005304111 0.6173340         0.6173340    -1.2152565     1.2046483
5 -0.005304111 0.6608347         0.6608347    -1.3005162     1.2899080

The forecasted plot is based on the best model ARIMAX(1,1,1)+GARCH(1,1). This model takes into account the autoregressive and moving average components of the data, as well as the impact off exogenous variables on the time series. Additionally, the GARCH component of the model accounts for the volatility clustering in the data. Overall, this model is well-suited to make accurate predictions about future values of the time series.

Equation of the Model

The equation of the ARIMAX(1,1,1) model is:

\(Y(t) = c + \phi_1(Y{(t-1)} - X{(t-1)}) + \theta_1\epsilon{(t-1)} + \epsilon(t)\)

where, \(Y(t)\) is the time series variable, \(X(t-1)\) is the exogenous variable, \(c\) is a constant, \(\phi_1\) and \(\theta_1\) are the parameters, and \(\epsilon(t)\) is the error term.

The equation of the GARCH(1,1) model is:

\(\sigma^2(t) = \alpha_0+\alpha_1\epsilon_t^2(t-1)+ \beta_1\sigma^2(t-1)\)

where \(\sigma^2_t\) is the conditional variance at time \(t\), \(\alpha_0\) is a constant, \(\alpha_1\) and \(\beta_1\) are the parameters, and \(\epsilon_t\) is the error term.

The combined equation of the ARIMAX(1,1,1)+GARCH(1,1) model is:

\(Y(t) = c + \phi_1(Y(t-1) - X(t-1)) + \theta_1\epsilon(t-1) + \epsilon(t)\)

\(\epsilon(t) = \sigma(t) * \epsilon~(t)\)

\(\sigma^2(t) = \alpha_0+\alpha_1\epsilon_t^2(t-1)+ \beta_1\sigma^2(t-1)\)